Linear Algebra provides a valuable introduction to the basic theory of matrices and vector spaces. The book covers: matrices, vector spaces, bases, and dimension, inner products, bilinear and sesquilinear forms over vector spaces, linear transformations, eigenvalues and eigenvectors, diagonalization, and Jordan normal form, and fields and polynomials over fields. Abstract methods are illustrated with concrete examples, and more detailed examples highlight applications of linear algebra to analysis, geometry, differential equations, relativity and quantum mechanics. Rigorous without being unnecessarily abstract, this useful and concise guide to the subject will be important reading for all students in mathematics and related fields.